Leonardo F. Cavenaghi

The family of invariant Riemannian manifolds in the Wallach flag manifold $\mathrm{SU}(3)/{\mathrm{T}}^2$ is described by three parameters $(x,y,z)$ of positive real numbers. By restricting such a family of metrics in the tetrahedron $\mathcal{T}:=x+y+z=1$, in this paper, we describe all regions $\mathcal{R}\subset \mathcal{T}$ admitting metrics with curvature properties varying from positive sectional curvature to positive scalar curvature, including positive intermediate curvature notion's. We study the dynamics of such regions under the projected Ricci flow in the plane $(x,y)$, concluding sign curvature maintenance and escaping. In addition, we obtain some results for positive intermediate Ricci curvature for a path of metrics on fiber bundles over $\mathrm{SU}(3)/{\mathrm{T}}^2$, further studying its evolution under the Ricci flow on the base.

In this note, we show that the classical Wallach manifold $\mathrm{SU}(3)/{\mathrm{T}}^2$-admits metrics of positive intermediate Ricci curvature $({\mathrm{Ric}}_d>0)$ for $d=1,2,3,4,5$ that lose these properties under the homogeneous Ricci flow for $d=1,2,3,5$. We make the same analyses to the family of Riemannian flag manifolds $\mathrm{SU}(m+2p)/\mathrm{S}(\mathrm{U}(m)\times \mathrm{U}(p)\times \mathrm{U}(p))$, concluding similar results. These explicitly verify some claims expected to be true among experts (see) for positive Ricci curvature and intermediate positive Ricci curvature. Our technique is only possible due to the global behavior understanding of the homogeneous Ricci flow for invariant metrics on these manifolds.